A general set of methods is presented for calculating chemical potentials in solid and liquid mixtures using {em ab initio} techniques based on density functional theory (DFT). The methods are designed to give an {em ab initio} approach to treating chemical equilibrium between coexisting solid and liquid solutions, and particularly the partitioning ratio of solutes between such solutions. For the liquid phase, the methods are based on the general technique of thermodynamic integration, applied to calculate the change of free energy associated with the continuous interconversion of solvent and solute atoms, the required thermal averages being computed by DFT molecular dynamics simulation. For the solid phase, free energies and hence chemical potentials are obtained using DFT calculation of vibrational frequencies of systems containing substitutional solute atoms, with anharmonic contributions calculated, where needed, by thermodynamic integration. The practical use of the methods is illustrated by applying them to study chemical equilibrium between the outer liquid and inner solid parts of the Earths core, modelled as solutions of S, Si and O in Fe. The calculations place strong constraints on the chemical composition of the core, and allow an estimate of the temperature at the inner-core/outer-core boundary.
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